Quasisymmetric Divided Differences and Forest Polynomials

Postnikov's divided symmetrization, introduced in the context of volume polynomials of permutahedra, possesses a host of remarkable ``positivity'' properties. These turn out to be best understood using a family of operators we call quasisymmetric divided differences.

I will introduce these operators and then define a basis of the polynomial ring adapted to these operators in the same way as ordinary divided differences interact with Schubert polynomials. This basis works nicely with respect to reduction modulo the ideal of positive degree quasisymmetric polynomials. Furthermore the expansion of the Schubert polynomials in this basis is nonnegative-- in fact it encodes the Schubert class expansions of the classes of certain toric Richardson varieties whose moment polytopes come from a cubical subdivision of the permutahedron. I will give a combinatorial procedure to compute these Schubert structure constants, and conclude by briefly revisiting mixed Eulerian numbers and lattice point counts of permutahedra

The underlying geometry is the subject of a follow-up talk by Hunter Spink.

Joint work with Philippe Nadeau (Lyon) and Hunter Spink (Toronto).